The evaluation of American option prices under stochastic volatility and jump-diffusion dynamics Carl Chiarella ∗ , Boda Kang ∗ , Gunter Meyer † and Andrew Ziogas ‡ ∗ School of Finance and Economics, UTS † School of Mathematics, Georgia Institute of Technology, Atlanta ‡ Integral Energy, Australia Workshop on Mathematical Finance Wolfgang Runggaldier’s 65th Birthday Bressanone 16-20 July, 2007
1 Introduction • Implied volatilities found using traded option prices vary with respect to option moneyness; smiles and skews are observed . • Use alternative to the Black-Scholes asset return dynamics to capture the leptokurtosis found in financial time series data e.g. Merton’s (1976) jump-diffusion model, Heston’s (1993) stochastic volatility model, Scott’s (1997) SV + Jumps model, L´ evy processes - Cont & Tankov (2004). • Pricing European options under these alternative dynamics well-developed. American option prices are much harder to evaluate in these cases. 1 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
• Aims of this paper are: – to derive the integral equations for the price and early exercise boundary of an American call option under the combination of Heston’s (1993) square root and Merton’s (1976) jump diffusion processes; – to extend the Fourier transform method as reformulated by Jamshid- ian (1992) to this; and – to investigate numerical solution of the IPDE via the method of lines . 2 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
2 Presentation Overview • Review of existing literature. • Problem definition. • The solution using the Fourier transform. • Structure of the solution. • Solving the IPDE directly via the method of lines. • Solving the IPDE via componentwise splitting. • Numerical results. • Conclusion. 3 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
3 Literature Review: American Option • Kim (1990), Jacka (1991), Carr, Jarrow & Myneni (1992) : various derivation methods; decompose option price into European price plus an early premium term. • Jamshidian (1992) : transforms homogeneous FBVP to inhomogeneous unrestricted BVP. • Meyer and Van der Hoek (1997) : Method of Lines. • Various authors : Finite difference methods, finite element methods etc. 4 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
4 Literature Review: Stochastic Volatility • Characteristic Functions (Heston, 1993): extended to American op- tions by Tzavalis & Wang (2003) . • Probability Methods (Jacka, 1991): extended to stochastic volatility for American options by Touzi (1999) . • Compound Option Approach (Kim, 1990; Geske & Johnson, 1984): extended by Zhylyevsky (2005) using Fast Fourier Transforms. • Incomplete Fourier Transform and modifications (McKean ,1965; Jamshidian, 1992): extended to stochastic volatility by Chiarella and Ziogas (2006) . 5 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
5 Literature Review: Jump Diffusions • Merton (1976) : European call options under jump-diffusion. • Pham (1997) : American puts under jump-diffusion; behaviour of the price and free boundary using Ito calculus. • Gukhal (2001) : Kim’s method for American calls and puts under jump- diffusion. • Meyer (1998) : Method of lines. 6 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
6 Literature Review: SV + JD • Bates (1996) : SV + JD model applied to Deutshe Mark options. • Scott (1997) : European options - Fourier transform approach. • Cont and Tankov (2004) : From the perspective of Levy processes. 7 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
7 SV + JD Dynamics • SDE for S : dS = ( µ − λk ) Sdt + √ vSdZ 1 + ( Y − 1) Sd ¯ q, dv = κ v ( θ − v ) dt + σ √ vdZ 2 , dZ j ∼ N (0 , dt ) , E [ dZ 1 dZ 2 ] = ρdt, � ∞ k = E Q [( Y − 1)] = ( Y − 1) G ( Y ) dY. 0 8 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
8 American Call: Free Boundary Value Problem � � Z ∞ • IPDE for C ( S, v, τ ) : ∂τ = vS 2 ∂ 2 C ∂S 2 + ρσvS ∂ 2 C ∂S∂v + σ 2 v ∂ 2 C ∂C ∂v 2 2 2 Z ∞ S ∂C + (1 − λ J ( Y ))( Y − 1) G ( Y ) dY r − q − λ ∂S 0 + ( κ v [ θ − v ] − λ v v ) ∂C ∂v − rC + λ (1 − λ J ( Y ))[ C ( SY, v, τ ) − C ( S, v, τ )] G ( Y ) dY, 0 • λ J ( Y ) - the MPR associated with a jump in S with magnitude Y . • Assume the market price of volatility risk is of the form √ v . λ ( v ) = λ v 9 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
• Solved in the region 0 ≤ τ ≤ T , 0 < S ≤ b ( v, τ ) . • Subject to BCs C ( S, v, 0) = max( S − K, 0) , C ( b ( v, τ ) , v, τ ) = b ( v, τ ) − K, ∂C ∂C lim ∂S = 1 , lim ∂v = 0 . S → b ( v,τ ) S → b ( v,τ ) 10 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
• In the volatility domain the boundary conditions are ∂C ( S, 0 , τ ) = ∂τ � ∞ � � S ∂C ( S, 0 , τ ) r − q − λ (1 − λ J ( Y ))( Y − 1) G ( Y ) dY ∂S 0 + κ v θ∂C ( S, 0 , τ ) − rC ( S, 0 , τ ) ∂v � ∞ + λ (1 − λ J ( Y ))[ C ( SY, 0 , τ ) − C ( S, 0 , τ )] G ( Y ) dY, 0 v →∞ C ( S, v, τ ) = S. lim 11 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
9 The Different Approaches to American Option Pricing M C Kean:- Domain Charge Jamshidiam:- Kim:- ✛ ✲ Homogeneous PDE Inhomogeneous PDE Compound Option 0 < S < a ( τ ) 0 < S < ∞ R τ R τ � � � � Incomplete Fourier Induction Fourier Transform Limit R τ Transform R τ ❄ ❄ ❄ � � � � C Am ( S,τ )= F M 0 G M ( a ( ξ ) , a ′ ( ξ ) ,S,ξ ) dξ ✛ ✲ C Am ( S,τ )= F K 0 G K ( a ( ξ ) ,S,ξ ) dξ Integration a ( τ )= F a 0 G a a ( τ )= F a 0 G a M ( a ( ξ ) , a ′ ( ξ ) ,a ( τ ) ,ξ ) dξ by Parts K ( a ( ξ ) ,a ( τ ) ,ξ ) dξ M K 12 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
C ( S, v, τ ) � C ( S − , v, τ ) • C ( S + , v, τ ) • • • K b ( v, τ ) S − S Cost S + = Y S − Cost incurred by the investor from downward jumps in S . 16 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
13 General Form of the Solution • The Jamshidiam approach gives the solution of the form � τ C ( S, v, τ ) = Ω C ( S, v, τ ) + Ψ C [ b ( ξ ) , ξ, τ, S, v ; C ( · , ξ )] dξ, 0 � τ b ( v, τ ) = Ω a ( b ( τ ) , v, τ ) + Ψ a [ b ( ξ ) , ξ, τ, b ( τ ) , v ; C ( · , ξ )] dξ, 0 31 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
14 Numerical Solution Using the Method of Lines • The method of lines has several strengths when dealing with American options: – The price , free boundary , delta and gamma are all found as part of the computation. – The method discretises the IPDE in an intuitive manner, and is read- ily adapted to be second order accurate in time . • The key idea behind the method of lines is to replace an IPDE with an equivalent system of one-dimensional integro-differential equations (IDEs). • The system of IDEs is developed by discretising the time derivative and the derivative terms involving the volatility, v . • We must provide a means of dealing with the integral term. 32 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
• The IPDE to be solved is ∂τ = vS 2 ∂ 2 C ∂S 2 + ρσvS ∂ 2 C ∂S∂v + σ 2 v ∂ 2 C ∂C ∂v 2 2 2 + ( r − q − λ ∗ k ∗ ) S ∂C ∂S + ( α − βv ) ∂C ∂v � ∞ − ( r + λ ∗ ) C + λ ∗ C ( SY, v, τ ) G ∗ ( Y ) dY, 0 where α ≡ κ v θ and β ≡ κ v + λ v . • The domain for the problem is 0 ≤ τ ≤ T , 0 ≤ S ≤ b ( v, τ ) and 0 ≤ v < ∞ . 33 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
• We discretise according to τ n = n ∆ τ and v m = m ∆ v . • C ( S, v m , τ n ) = C n m ( S ) , V ( S, v m , τ n ) ≡ ∂C ( S, v m , τ n ) = V n m ( S ) . ∂S • We use the standard central difference scheme ∂ 2 C ∂v 2 = C n m +1 − 2 C n m + C n m − 1 , (∆ v ) 2 ∂S∂v = V n ∂ 2 C m +1 − V n m − 1 . 2∆ v • We use an upwinding finite difference scheme for the first order deriv- ative term C n m +1 − C n v ≤ α m β , if ∂C ∆ v ∂v = C n m − C n v > α m − 1 β . if ∆ v 34 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
• Integral term at each grid point estimated via num. integration. • We assume that the jump sizes are log-normally distributed. • Applying the Hermite Gauss-quadrature scheme, we have J √ 1 � � �� � ( γ − δ 2 / 2) + W n w H j C n 2 δX H m = S exp , √ π m j j =0 • We interpolate for the required values of C n m using cubic splines fitted in S along each line in v . • A second order approximation for the time derivative , m − C n − 1 C n − 1 − C n − 2 C n ∂C ∂τ = 3 + 1 m m m . 2 ∆ τ 2 ∆ τ 35 Chiarella, Kang, Meyer and Ziogas American Options – Stochastic Volatility + Jumps
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